Korteweg–De Vries Equation - Asymptotic Decomposition into Solitons

Описание: Source code available at: https://github.com/RichtersFinger/pse...


The Korteweg–De Vries (KdV) equation [1] is a simple, spatially one-dimensional model for the evolution of solitary waves [2,3]. Its dynamics can be observed for example in the context of shallow-water waves.


A solitary wave in the KdV equation is characterized by having a specific fixed propagation velocity and a directly related constant shape. Furthermore, a soliton recovers its properties after a collision with another soliton. A common observation is how an initial condition that is not a solitary wave is quickly separated into a superposition of solitary waves.
The visualization below illustrates this phenomenon for three different initial conditions.


[1] Su, Chau Hsing, and Clifford S. Gardner, "Korteweg‐de Vries equation and generalizations. III. Derivation of the Korteweg‐de Vries equation and Burgers equation", Journal of Mathematical Physics 10.3, 536 (1969).
[2] Hansen, Paul J., and Dwight Roy Nicholson, "Simple soliton solutions", American Journal of Physics 47.9, 769 (1979).
[3] Taylor, Beverley AP, "What is a solitary wave?", American Journal of Physics 47.10, 847 (1979).